from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1849, base_ring=CyclotomicField(1806))
M = H._module
chi = DirichletCharacter(H, M([2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(9,1849))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1849\) | |
| Conductor: | \(1849\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(903\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{903})$ |
| Fixed field: | Number field defined by a degree 903 polynomial (not computed) |
First 31 of 504 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1849}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{212}{301}\right)\) | \(e\left(\frac{1}{903}\right)\) | \(e\left(\frac{123}{301}\right)\) | \(e\left(\frac{193}{903}\right)\) | \(e\left(\frac{91}{129}\right)\) | \(e\left(\frac{35}{129}\right)\) | \(e\left(\frac{34}{301}\right)\) | \(e\left(\frac{2}{903}\right)\) | \(e\left(\frac{829}{903}\right)\) | \(e\left(\frac{185}{301}\right)\) |
| \(\chi_{1849}(10,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{283}{301}\right)\) | \(e\left(\frac{866}{903}\right)\) | \(e\left(\frac{265}{301}\right)\) | \(e\left(\frac{83}{903}\right)\) | \(e\left(\frac{116}{129}\right)\) | \(e\left(\frac{124}{129}\right)\) | \(e\left(\frac{247}{301}\right)\) | \(e\left(\frac{829}{903}\right)\) | \(e\left(\frac{29}{903}\right)\) | \(e\left(\frac{78}{301}\right)\) |
| \(\chi_{1849}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{74}{301}\right)\) | \(e\left(\frac{520}{903}\right)\) | \(e\left(\frac{148}{301}\right)\) | \(e\left(\frac{127}{903}\right)\) | \(e\left(\frac{106}{129}\right)\) | \(e\left(\frac{11}{129}\right)\) | \(e\left(\frac{222}{301}\right)\) | \(e\left(\frac{137}{903}\right)\) | \(e\left(\frac{349}{903}\right)\) | \(e\left(\frac{181}{301}\right)\) |
| \(\chi_{1849}(14,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{76}{301}\right)\) | \(e\left(\frac{892}{903}\right)\) | \(e\left(\frac{152}{301}\right)\) | \(e\left(\frac{586}{903}\right)\) | \(e\left(\frac{31}{129}\right)\) | \(e\left(\frac{2}{129}\right)\) | \(e\left(\frac{228}{301}\right)\) | \(e\left(\frac{881}{903}\right)\) | \(e\left(\frac{814}{903}\right)\) | \(e\left(\frac{72}{301}\right)\) |
| \(\chi_{1849}(15,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{96}{301}\right)\) | \(e\left(\frac{97}{903}\right)\) | \(e\left(\frac{192}{301}\right)\) | \(e\left(\frac{661}{903}\right)\) | \(e\left(\frac{55}{129}\right)\) | \(e\left(\frac{41}{129}\right)\) | \(e\left(\frac{288}{301}\right)\) | \(e\left(\frac{194}{903}\right)\) | \(e\left(\frac{46}{903}\right)\) | \(e\left(\frac{186}{301}\right)\) |
| \(\chi_{1849}(17,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{301}\right)\) | \(e\left(\frac{334}{903}\right)\) | \(e\left(\frac{146}{301}\right)\) | \(e\left(\frac{349}{903}\right)\) | \(e\left(\frac{79}{129}\right)\) | \(e\left(\frac{80}{129}\right)\) | \(e\left(\frac{219}{301}\right)\) | \(e\left(\frac{668}{903}\right)\) | \(e\left(\frac{568}{903}\right)\) | \(e\left(\frac{85}{301}\right)\) |
| \(\chi_{1849}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{254}{301}\right)\) | \(e\left(\frac{890}{903}\right)\) | \(e\left(\frac{207}{301}\right)\) | \(e\left(\frac{200}{903}\right)\) | \(e\left(\frac{107}{129}\right)\) | \(e\left(\frac{61}{129}\right)\) | \(e\left(\frac{160}{301}\right)\) | \(e\left(\frac{877}{903}\right)\) | \(e\left(\frac{59}{903}\right)\) | \(e\left(\frac{3}{301}\right)\) |
| \(\chi_{1849}(24,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{82}{301}\right)\) | \(e\left(\frac{503}{903}\right)\) | \(e\left(\frac{164}{301}\right)\) | \(e\left(\frac{458}{903}\right)\) | \(e\left(\frac{107}{129}\right)\) | \(e\left(\frac{61}{129}\right)\) | \(e\left(\frac{246}{301}\right)\) | \(e\left(\frac{103}{903}\right)\) | \(e\left(\frac{704}{903}\right)\) | \(e\left(\frac{46}{301}\right)\) |
| \(\chi_{1849}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{281}{301}\right)\) | \(e\left(\frac{193}{903}\right)\) | \(e\left(\frac{261}{301}\right)\) | \(e\left(\frac{226}{903}\right)\) | \(e\left(\frac{19}{129}\right)\) | \(e\left(\frac{47}{129}\right)\) | \(e\left(\frac{241}{301}\right)\) | \(e\left(\frac{386}{903}\right)\) | \(e\left(\frac{166}{903}\right)\) | \(e\left(\frac{187}{301}\right)\) |
| \(\chi_{1849}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{301}\right)\) | \(e\left(\frac{899}{903}\right)\) | \(e\left(\frac{110}{301}\right)\) | \(e\left(\frac{131}{903}\right)\) | \(e\left(\frac{23}{129}\right)\) | \(e\left(\frac{118}{129}\right)\) | \(e\left(\frac{165}{301}\right)\) | \(e\left(\frac{895}{903}\right)\) | \(e\left(\frac{296}{903}\right)\) | \(e\left(\frac{163}{301}\right)\) |
| \(\chi_{1849}(38,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{207}{301}\right)\) | \(e\left(\frac{275}{903}\right)\) | \(e\left(\frac{113}{301}\right)\) | \(e\left(\frac{701}{903}\right)\) | \(e\left(\frac{128}{129}\right)\) | \(e\left(\frac{79}{129}\right)\) | \(e\left(\frac{19}{301}\right)\) | \(e\left(\frac{550}{903}\right)\) | \(e\left(\frac{419}{903}\right)\) | \(e\left(\frac{6}{301}\right)\) |
| \(\chi_{1849}(40,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{267}{301}\right)\) | \(e\left(\frac{599}{903}\right)\) | \(e\left(\frac{233}{301}\right)\) | \(e\left(\frac{23}{903}\right)\) | \(e\left(\frac{71}{129}\right)\) | \(e\left(\frac{67}{129}\right)\) | \(e\left(\frac{199}{301}\right)\) | \(e\left(\frac{295}{903}\right)\) | \(e\left(\frac{824}{903}\right)\) | \(e\left(\frac{47}{301}\right)\) |
| \(\chi_{1849}(52,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{301}\right)\) | \(e\left(\frac{253}{903}\right)\) | \(e\left(\frac{116}{301}\right)\) | \(e\left(\frac{67}{903}\right)\) | \(e\left(\frac{61}{129}\right)\) | \(e\left(\frac{83}{129}\right)\) | \(e\left(\frac{174}{301}\right)\) | \(e\left(\frac{506}{903}\right)\) | \(e\left(\frac{241}{903}\right)\) | \(e\left(\frac{150}{301}\right)\) |
| \(\chi_{1849}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{301}\right)\) | \(e\left(\frac{551}{903}\right)\) | \(e\left(\frac{48}{301}\right)\) | \(e\left(\frac{692}{903}\right)\) | \(e\left(\frac{89}{129}\right)\) | \(e\left(\frac{64}{129}\right)\) | \(e\left(\frac{72}{301}\right)\) | \(e\left(\frac{199}{903}\right)\) | \(e\left(\frac{764}{903}\right)\) | \(e\left(\frac{197}{301}\right)\) |
| \(\chi_{1849}(56,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{60}{301}\right)\) | \(e\left(\frac{625}{903}\right)\) | \(e\left(\frac{120}{301}\right)\) | \(e\left(\frac{526}{903}\right)\) | \(e\left(\frac{115}{129}\right)\) | \(e\left(\frac{74}{129}\right)\) | \(e\left(\frac{180}{301}\right)\) | \(e\left(\frac{347}{903}\right)\) | \(e\left(\frac{706}{903}\right)\) | \(e\left(\frac{41}{301}\right)\) |
| \(\chi_{1849}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{301}\right)\) | \(e\left(\frac{409}{903}\right)\) | \(e\left(\frac{40}{301}\right)\) | \(e\left(\frac{376}{903}\right)\) | \(e\left(\frac{67}{129}\right)\) | \(e\left(\frac{125}{129}\right)\) | \(e\left(\frac{60}{301}\right)\) | \(e\left(\frac{818}{903}\right)\) | \(e\left(\frac{436}{903}\right)\) | \(e\left(\frac{114}{301}\right)\) |
| \(\chi_{1849}(58,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{124}{301}\right)\) | \(e\left(\frac{790}{903}\right)\) | \(e\left(\frac{248}{301}\right)\) | \(e\left(\frac{766}{903}\right)\) | \(e\left(\frac{37}{129}\right)\) | \(e\left(\frac{44}{129}\right)\) | \(e\left(\frac{71}{301}\right)\) | \(e\left(\frac{677}{903}\right)\) | \(e\left(\frac{235}{903}\right)\) | \(e\left(\frac{165}{301}\right)\) |
| \(\chi_{1849}(60,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{80}{301}\right)\) | \(e\left(\frac{733}{903}\right)\) | \(e\left(\frac{160}{301}\right)\) | \(e\left(\frac{601}{903}\right)\) | \(e\left(\frac{10}{129}\right)\) | \(e\left(\frac{113}{129}\right)\) | \(e\left(\frac{240}{301}\right)\) | \(e\left(\frac{563}{903}\right)\) | \(e\left(\frac{841}{903}\right)\) | \(e\left(\frac{155}{301}\right)\) |
| \(\chi_{1849}(66,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{233}{301}\right)\) | \(e\left(\frac{596}{903}\right)\) | \(e\left(\frac{165}{301}\right)\) | \(e\left(\frac{347}{903}\right)\) | \(e\left(\frac{56}{129}\right)\) | \(e\left(\frac{91}{129}\right)\) | \(e\left(\frac{97}{301}\right)\) | \(e\left(\frac{289}{903}\right)\) | \(e\left(\frac{143}{903}\right)\) | \(e\left(\frac{94}{301}\right)\) |
| \(\chi_{1849}(67,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{250}{301}\right)\) | \(e\left(\frac{146}{903}\right)\) | \(e\left(\frac{199}{301}\right)\) | \(e\left(\frac{185}{903}\right)\) | \(e\left(\frac{128}{129}\right)\) | \(e\left(\frac{79}{129}\right)\) | \(e\left(\frac{148}{301}\right)\) | \(e\left(\frac{292}{903}\right)\) | \(e\left(\frac{32}{903}\right)\) | \(e\left(\frac{221}{301}\right)\) |
| \(\chi_{1849}(68,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{301}\right)\) | \(e\left(\frac{67}{903}\right)\) | \(e\left(\frac{114}{301}\right)\) | \(e\left(\frac{289}{903}\right)\) | \(e\left(\frac{34}{129}\right)\) | \(e\left(\frac{23}{129}\right)\) | \(e\left(\frac{171}{301}\right)\) | \(e\left(\frac{134}{903}\right)\) | \(e\left(\frac{460}{903}\right)\) | \(e\left(\frac{54}{301}\right)\) |
| \(\chi_{1849}(74,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{301}\right)\) | \(e\left(\frac{710}{903}\right)\) | \(e\left(\frac{40}{301}\right)\) | \(e\left(\frac{677}{903}\right)\) | \(e\left(\frac{110}{129}\right)\) | \(e\left(\frac{82}{129}\right)\) | \(e\left(\frac{60}{301}\right)\) | \(e\left(\frac{517}{903}\right)\) | \(e\left(\frac{737}{903}\right)\) | \(e\left(\frac{114}{301}\right)\) |
| \(\chi_{1849}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{123}{301}\right)\) | \(e\left(\frac{2}{903}\right)\) | \(e\left(\frac{246}{301}\right)\) | \(e\left(\frac{386}{903}\right)\) | \(e\left(\frac{53}{129}\right)\) | \(e\left(\frac{70}{129}\right)\) | \(e\left(\frac{68}{301}\right)\) | \(e\left(\frac{4}{903}\right)\) | \(e\left(\frac{755}{903}\right)\) | \(e\left(\frac{69}{301}\right)\) |
| \(\chi_{1849}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{301}\right)\) | \(e\left(\frac{746}{903}\right)\) | \(e\left(\frac{254}{301}\right)\) | \(e\left(\frac{401}{903}\right)\) | \(e\left(\frac{32}{129}\right)\) | \(e\left(\frac{52}{129}\right)\) | \(e\left(\frac{80}{301}\right)\) | \(e\left(\frac{589}{903}\right)\) | \(e\left(\frac{782}{903}\right)\) | \(e\left(\frac{152}{301}\right)\) |
| \(\chi_{1849}(95,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{205}{301}\right)\) | \(e\left(\frac{505}{903}\right)\) | \(e\left(\frac{109}{301}\right)\) | \(e\left(\frac{844}{903}\right)\) | \(e\left(\frac{31}{129}\right)\) | \(e\left(\frac{2}{129}\right)\) | \(e\left(\frac{13}{301}\right)\) | \(e\left(\frac{107}{903}\right)\) | \(e\left(\frac{556}{903}\right)\) | \(e\left(\frac{115}{301}\right)\) |
| \(\chi_{1849}(96,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{66}{301}\right)\) | \(e\left(\frac{236}{903}\right)\) | \(e\left(\frac{132}{301}\right)\) | \(e\left(\frac{398}{903}\right)\) | \(e\left(\frac{62}{129}\right)\) | \(e\left(\frac{4}{129}\right)\) | \(e\left(\frac{198}{301}\right)\) | \(e\left(\frac{472}{903}\right)\) | \(e\left(\frac{596}{903}\right)\) | \(e\left(\frac{15}{301}\right)\) |
| \(\chi_{1849}(99,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{301}\right)\) | \(e\left(\frac{730}{903}\right)\) | \(e\left(\frac{92}{301}\right)\) | \(e\left(\frac{22}{903}\right)\) | \(e\left(\frac{124}{129}\right)\) | \(e\left(\frac{8}{129}\right)\) | \(e\left(\frac{138}{301}\right)\) | \(e\left(\frac{557}{903}\right)\) | \(e\left(\frac{160}{903}\right)\) | \(e\left(\frac{202}{301}\right)\) |
| \(\chi_{1849}(100,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{265}{301}\right)\) | \(e\left(\frac{829}{903}\right)\) | \(e\left(\frac{229}{301}\right)\) | \(e\left(\frac{166}{903}\right)\) | \(e\left(\frac{103}{129}\right)\) | \(e\left(\frac{119}{129}\right)\) | \(e\left(\frac{193}{301}\right)\) | \(e\left(\frac{755}{903}\right)\) | \(e\left(\frac{58}{903}\right)\) | \(e\left(\frac{156}{301}\right)\) |
| \(\chi_{1849}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{152}{301}\right)\) | \(e\left(\frac{580}{903}\right)\) | \(e\left(\frac{3}{301}\right)\) | \(e\left(\frac{871}{903}\right)\) | \(e\left(\frac{19}{129}\right)\) | \(e\left(\frac{47}{129}\right)\) | \(e\left(\frac{155}{301}\right)\) | \(e\left(\frac{257}{903}\right)\) | \(e\left(\frac{424}{903}\right)\) | \(e\left(\frac{144}{301}\right)\) |
| \(\chi_{1849}(103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{87}{301}\right)\) | \(e\left(\frac{229}{903}\right)\) | \(e\left(\frac{174}{301}\right)\) | \(e\left(\frac{853}{903}\right)\) | \(e\left(\frac{70}{129}\right)\) | \(e\left(\frac{17}{129}\right)\) | \(e\left(\frac{261}{301}\right)\) | \(e\left(\frac{458}{903}\right)\) | \(e\left(\frac{211}{903}\right)\) | \(e\left(\frac{225}{301}\right)\) |
| \(\chi_{1849}(109,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{212}{301}\right)\) | \(e\left(\frac{302}{903}\right)\) | \(e\left(\frac{123}{301}\right)\) | \(e\left(\frac{494}{903}\right)\) | \(e\left(\frac{5}{129}\right)\) | \(e\left(\frac{121}{129}\right)\) | \(e\left(\frac{34}{301}\right)\) | \(e\left(\frac{604}{903}\right)\) | \(e\left(\frac{227}{903}\right)\) | \(e\left(\frac{185}{301}\right)\) |